Conservation of momentum – walking the boat
Akio Saitoh
Department of Civil Engineering, Kinki University, 8-23-1,
Shinke-cho, Yao-shi, Osaka 581-0811, Japan
E-mail: [email protected]
(Received 18 August 2010; accepted 18 September 2010)
Abstract
We present an apparatus for checking the law of conservation of momentum in the student lab. We can easily construct
it at a low cost and it yields good results.
Keywords: Conservation of momentum, Frictional force.
Resumen
Se presenta un aparato para revisar la ley de conservación del momento en el laboratorio para estudiantes. Es fácil de
construir a bajo costo y produce buenos resultados.
Palabras clave: Conservación de momento, Fuerza de fricción.
PACS: 45.20.D-, 45.20.df, 45.50.Tn
ISSN 1870-9095
I. INTRODUCCIÓN
A man of mass 60 kg is standing on a flat boat at a point
10 m from shore as shown in Fig.1.
Many texts [1, 2, 3] contain problems similar to the
following:
FIGURE 1. If a man walks on a boat toward shore, the boat moves in opposite direction.
He walks 2.4 m relative to the boat toward shore and then
stops. If the mass of the boat is 300 kg and it is assumed
there is no friction between the boat and the water, how far
is the man from shore when he stops?
Conservation of momentum predicts that the center of
mass of the boat-man system does not move. Application of
this concept predicts that, while the man walked 2.4 m on
the boat, the boat moves away from shore 0.4 m and the
man is now 8 m from shore.
Lat. Am. J. Phys. Educ. Vol. 4,No. 3, Sept. 2010
The apparatus described here allows checking the above
calculation in the student laboratory. Construction cost is
low and the results agree well with theory.
Figure 2 shows a runway made of acrylic sheet which
can roll freely over another acrylic plate separated from it
by many steel balls of diameter 0.475 cm, which greatly
reduce any friction. The lower plate must be carefully
leveled.
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Conservation of momentum – walking the boat
FIGURA 2. An apparatus for checking conservation of momentum.
A spring-wound toy car placed at one end of the runway is
released and the car and runway begin to move in opposite
directions. When the car strikes a bumper at the end, both
car and runway stop simultaneously. The ratio of the
distances moved by car and runway as well as their
respective velocities, relative to the fixed lower plate, are in
the inverse ratio of their masses.
Using this apparatus, the two experiments shown in Fig.
3 were performed.
FIGURA 3. (a) The car moves 24 cm, and the runway moves 12 cm. (b) Both car and runway move the same distance, 18 cm.
Lat. Am. J. Phys. Educ. Vol. 4,No. 3, Sept. 2010
773
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Akio Saitoh
In Fig. 3a, the car has a mass of 90 g and moves 24 cm
while the runway has a mass of 180 g and should move 12
cm in the opposite direction. In Fig. 3b, the mass of car and
runway are 180 g each and each should move 18 cm.
Experimental results were in good agreement with these
predictions.
An analysis of the role of the frictional forces between
the runway are the lower plate reveals the reasons for the
validity of the assumption that friction is negligible. Figure
4 is the side view of the car, runway, and lower plate,
showing the forces on each.
FIGURA 4. The side view of the apparatus. Arrow shows a force and double-arrow shows an acceleration.
where a is the acceleration of the ball. The rotational
motion of the ball about their center of mass follows from
If the spring in the car provides a force F of the wheels on
the runway which is assumed to be constant over these short
travel distances, then a constant acceleration a results and
the distance S traveled by the car can be found from
(f
f )r
I ,
where I is the moment of inertia,
(3)
2
2
m r , and α is the
5
S
1
at
2
2
F
2
t .
angular acceleration,
(1)
2m
f
ma,
Similarly, for the runway of mass
r
M,
where t is the time the car is in motion on the runway and
m is the mass of the car.
There are 30 balls of mass m and radius r . In Fig. 4,
f and f are the forces of friction between runway and ball,
and acrylic plate and ball respectively. These forces act
horizontally, tangent to the ball, at the points of contact, P
and Q, as shown. It is assumed that the forces producing
translational motion of the ball act at the center of mass
such that
f
a
F
nf
MA.
(4)
Where n is the number of balls. The acceleration of the
runway is
A
2r .
(5)
The distance moved by the runway can be found using
(2)
X
1
At
2
(6)
.
2
Combining Eqs. (1), (4), and (6), we obtain
Lat. Am. J. Phys. Educ. Vol. 4,No. 3, Sept. 2010
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Conservation of momentum – walking the boat
S
M
X
m
F
F
.
Using the above numerical values, nf
(7)
nf
Eq. (7)
(8)
0.7 m r
nf
which can be combined to get
nf
F
7 nm
20 M
.
M
[1] Gambhir, D., IntroductoryPhysics Vol.1 (Mc Graw-Hill,
USA, 1976), p.35.
[2] Shortley, G. and Williams, D., Principles of College
Physics (Prentice-Hall, USA, 1959), p.160.
[3] White, H. E., Modern College Physics, 3rd ed. (Van
NostRand, USA, 1956), p.137.
(9)
2Mr
1.03
REFRENCES
and from Eqs. (4) and (5)
F
S
which explains the excellent
X
m
experimental result.
Using the apparatus described above, the experimental
results were in good agreement with the law of conservation
of momentum.
Equation (7) shows that the rolling friction of the balls can
be neglected if the value of nf is very small compared to F .
Let us estimate the ratio of nf and F for this arrangement
using n=30, mˈ=0.441g, and M=180g. From Eqs. (2) and
(3)
f
0.025 . Then, from
F
(10)
7 nm
Lat. Am. J. Phys. Educ. Vol. 4,No. 3, Sept. 2010
775
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